You are not wrong.
If you're birthday is transposed you will never be able to get a SSA account to find out what it should be. I have been fighting to get mine fixed for over 4 years now. I did a whole new citizenship form at the SSA "to renter it all correctly" and ended up losing my citizenship and almost got deported getting a drivers license. This was right during the early political travel ban, and I got scooped up by ICE at the DMV after trying to fix this. Try explaining a date transposition error to ICE for 7 hours, when they think all your docs are falsified.
I just file by mail now.
